Accelerators Lecture III

22
http://bruening.home.cern.ch/bruening/summer-school/lecture3 Accelerators Oliver Brüning SL/AP Lecture III

Transcript of Accelerators Lecture III

http://bruening.home.cern.ch/bruening/summer−school/lecture3

Accelerators

Oliver Brüning SL/AP

Lecture III

Longitudinal Focusing

Beam Optics

Weak and Strong Focusing

Summary Lecture II

Collider Concept

Long Term Stability

Chromaticity + Sextupoles

Closed Orbit

Linear Resonances Orbit Stability+

Long Term Stability

Summary

Orbit Stability +III)

Non-linear resonances

Detuning with amplitude

BDBFB

D

x = x + x0

dipole component

quadrupole

y

xB = -g yB = -g x - g x

orbit error

y

xB = -g y

0

B = -g x

B F B DB

F

Closed Orbit

Orbit Offset in Quadrupole:

calibration

power supplies

in Quadrupoles

slow driftcivilisationmoonseasonscivil engineering

Error in dipole strength

Sources for Orbit Offsets

Alignment: +/- 0.1 mm

Ground motion

Energy error of particles

rms < 0.5 mm x, y < 4 mm

2 300 correctors

2 540 monitors

784 quadrupoles

field imperfections

ernergy error

aperture

x-y coupling

beam monitors and orbit correctors

Aim:∆

Orbit error:

LEP:

t

γ

V

a)

b)

π

Equilibrium:RF

particle energy!

f = h frev

f = 1

energy increase

2q

mB

rev

E depends on orbit and magnetic field!

Synchrotron:

assume: L > design orbit

the orbit determines the

(s + L) = (s)

x (s + L) = x (s)

closed orbit

x (s); y (s)

oscillations around

find closed orbit

transverse

CO: periodic:common to all particles

periodic:common to all particles

φ, β: 0 0

00

β

of particle motion

β

beam ensemblebeam quality

ε:

Particle Motion:

motion0 individual particleφ , A:

0

complete description

0

Orbit Stability

Kick

watch out for integer tunes!

Q = N

the perturbation adds up

the perturbation cancels

Dipole Error and

after each turn

Kick

Q = N + 0.5

Q: βnumber of -oscillations per turn

Quadrupole Error:

amplitude increase

1. Turn: x > 0

Q = N + 0.5

F

kick

amplitude increase

beam offset in quadrupole

Fx < 02. Turn:

watch out for half integer tunes!

Orbit StabilityQuadrupole Error and

orbit kick proportional to

n n + 0.5 n + 1

Qx

n + 0.5

n + 1

Qy

Problem:

Q = Q +

K (quadrupole) = e gp (lecture II)

ξ ∆ p

requires correction!

p00

Large Machine (LEP / LHC):ξ 100 - 500

∆ pp0

10-3

Tune Diagram

N

S

N

S

N

2

S

coil

x

y

Orbit Offset:

B = g x y + g x y0

y = y + y0

B = g x y12B = g ( x - y )22

x

y

Sextupole Magnet

12

2 20B = g ( x - y ) - g y y

quadrupole component

[ g ] = T / m

pp > 0∆

0

pp∆

0

< 0

pp∆

0

pp∆

0

pp∆

0Q = Q + Q + Q

0

non-linear resonances

∆ ∆Q S

������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������������������������

Chromaticity Correction

SextupoleQuadrupole

offset in sextupole

x(s) = x (s) + D(s)0

0

Problem:

(β2 πL

φ0x = a A sin Q s + = const)

x = b A cos Q s +

x

x

Display coordinates after each turn:

x

x

= const.+ x 2

b2ax

2

2

ellipse

βLinear - motion:

Poincare Section

2 πL

φ0

= A

B = 21 g xy

2

Lorentz Force:

Fv px =

B yx = q p

Sextupole Perturbation

Sextupole Magnet:

x =∆

l

= 1v g x2

2l q

m

Poincare section: x

x

dsv pF

Sextupole Kick:

Q, 3 Q = n

x2 x 2

x x +2 x x2

φ π/ Turn = 2 Q)(∆

R

Amplitude Growth

Many Turns:

= qm v

l14 g

Turnφ φ3cos( ) + cos(3 )

∆ = 0 unless:

Perturbation Theory:

all resonances are driven!

R = +

ds= dR

∆ R

Detuning with Amplitude

Non-linear Perturbation:

Poincare section

Q

Q∆

Q

Q∆

avoid resonances:

Problem:

r Q = n !

there are resonances everywhere!

Stabilisation Mechanism:

∆x x 2

Complex dynamics:

no analytical solution!

analysis of long term stability

relies on numerical simulations

2

Long Term Stability

Non-linear Perturbation:

amplitude growth

detuning with amplitude

coupling

y

3 degrees of freedom

+ 1 invariant of the motion

non-linear dynamics+

sextupole: 2(B = g [x - y ])

Sources for Non-Linear Fields

components

Sextupoles

Magnet errors:

pole face accuracy

geometry errors

eddy currents

edge effects

Vacuum chamber:

LEP I welding

Beam-beam interaction

careful analysis of all

Chromaticity:

resonance driving terms

sextupoles

Non-Linear Resonances:

amplitude growth

detuning with amplitude

long term stability?

chaos theory

Linear Optics:1Q = n ; n +2

π π

Summary Resonances

classical mechanics +

������������������������������������������

������������������������������������������

������������������������������������������������������

������������������������������������������������������

MM

M

MM 123

4

5

One Turn Map (Taylor Series)

Hamilton Function

Accelerator Model

Toy Model:

HO perturbation

Hamilton Function# > 1000 elements!

+

simple

Element by Element Tracking

numberical analysis

Dyn

amic

Ape

rtur

e

Ham

ilton

Fun

ctio

nM

apM

odel

:

Per

turb

atio

n T

heor

y:L

inea

r +

ε*N

onlin

ear

Num

eric

al S

imul

atio

n:F

ast

Com

pute

r C

ode

Opt

ics

Clo

sed

Orb

it

Tun

e

Chr

omat

icit

y

Mul

tipo

leE

rror

sR

eson

ance

Stre

ngth

Det

unin

gD

etun

ing

Res

onan

ceSt

reng

th

Dyn

amic

Ape

rtur

e